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ArticleOriginal scientific text

Title

Ideals of finite rank operators, intersection properties of balls, and the approximation property

Authors 1, 2

Affiliations

  1. Department of Mathematics, Agder College, Tordenskjoldsgate 65, N-4604 Kristiansand, Norway
  2. Faculty of Mathematics, Tartu University, Vanemuise 46, EE-2400 Tartu, Estonia

Abstract

We characterize the approximation property of Banach spaces and their dual spaces by the position of finite rank operators in the space of compact operators. In particular, we show that a Banach space E has the approximation property if and only if for all closed subspaces F of c0, the space ℱ(F,E) of finite rank operators from F to E has the n-intersection property in the corresponding space K(F,E) of compact operators for all n, or equivalently, ℱ(F,E) is an ideal in K(F,E).

Bibliography

  1. N. Aronszajn and P. Panitchpakdi, Extension of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 6 (1956), 405-439.
  2. C.-M. Cho, The metric approximation property and intersection properties of balls, J. Korean Math. Soc. 31 (1994), 467-475.
  3. J. Diestel, Sequences and Series in Banach Spaces, Springer, New York, 1984.
  4. M. Feder and P. Saphar, Spaces of compact operators and their dual spaces, Israel J. Math. 21 (1975), 38-49.
  5. T. Figiel, Factorization of compact operators and applications to the approximation problem, Studia Math. 45 (1973), 191-210.
  6. G. Godefroy, N. J. Kalton and P. D. Saphar, Unconditional ideals in Banach spaces, ibid. 104 (1993), 13-59.
  7. A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955).
  8. P. Harmand, D. Werner and W. Werner, M-Ideals in Banach Spaces and Banach Algebras, Lecture Notes in Math. 1547, Springer, Berlin, 1993.
  9. H. Jarchow, Locally Convex Spaces, Teubner, Stuttgart, 1981.
  10. W. B. Johnson, Factoring compact operators, Israel J. Math. 9 (1971), 337-345.
  11. Å. Lima, Uniqueness of Hahn-Banach extensions and liftings of linear dependences, Math. Scand. 53 (1983), 97-113.
  12. Å. Lima, The metric approximation property, norm-one projections and intersection properties of balls, Israel J. Math. 84 (1993), 451-475.
  13. Å. Lima, Property (wM*) and the unconditional metric compact approximation property, Studia Math. 113 (1995), 249-263.
  14. J. Lindenstrauss, On a problem of Nachbin concerning extension of operators, Israel J. Math. 1 (1963), 75-84.
  15. J. Lindenstrauss, Extension of compact operators, Mem. Amer. Math. Soc. 48 (1964).
  16. J. Lindenstrauss, On projections with norm 1-an example, Proc. Amer. Math. Soc. 15 (1964), 403-406.
  17. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Ergeb. Math. Grenzgeb. 92, Springer, Berlin, 1977.
  18. E. Oja and M. Põldvere, On subspaces of Banach spaces where every functional has a unique norm-preserving extension, Studia Math. 117 (1996), 289-306.
  19. R. R. Phelps, Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Math. 1364, Springer, Berlin, 1993.
  20. W. M. Ruess and C. P. Stegall, Extreme points in duals of operator spaces, Math. Ann. 261 (1982), 535-546.
  21. I. Singer, Bases in Banach Spaces II, Springer, Berlin, 1981.
Studia Mathematica
1999/133/2
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Pages:
175-186
Main language of publication
English
Received
1998-05-11
Accepted
1998-08-24
Published
1999
Exact and natural sciences